Maths for Brain Imaging

This lecture series was given in the Autumn Term, 2006 at the Wellcome Trust Centre for Human Neuroimaging at UCL. It covers a number of mathematical methods that are used in the analysis of brain imaging data. Each lecture describes a different category of model and shows how it is applied to a particular aspect of brain imaging analysis. The applications cover data from functional Magnetic Resonance Imaging (fMRI), Magnetoencephalography (MEG) and Electroencephalography (EEG).

Download all course notes: PDF

1. General Linear Models I

    • Maximum likelihood estimation
    • Regression and correlation
    • Linear algebra
    • Functions of random vectors
    • Multiple regression and partial correlation
    • Application: fMRI time series

Lecture notes and code can be downloaded from this archive: ZIP

2. General Linear Models II

    • Estimating error variance
    • Comparing nested models
    • Transforming probability densities
    • Contrasts
    • Hemodynamic basis functions
    • Application: fMRI time series

Notes and matlab code can be downloaded here: ZIP

3. Random Field Theory

    • Gaussian processes
    • Covariance functions
    • Upcrossings of one-dimensional processes
    • Euler characteristic
    • Application: Detecting activations in fMRI data

Notes and matlab code for 1D/2D fields can be downloaded here: ZIP

4. Multivariate Models

    • More Linear Algebra
    • Principal component analysis
    • Singular Value Decomposition
    • Structural Equation Modelling
    • Granger causality
    • Application: PET & fMRI connectivity analyses

Notes and matlab code can be downloaded here: ZIP

5. Variance Components

    • GLMs with arbitrary error covariance
    • Weighted Least Squares
    • Restricted Maximum Likelihood
    • Application: fMRI time series analysis with correlated errors
    • Hierarchical Models
    • Application: Analysis of imaging data from a group

Notes and matlab code for ReML estimation of variance components can be downloaded here: ZIP


6. Bayesian methods

    • Bayes rule for Gaussians
    • Bayesian GLMs
    • Parametric Empirical Bayes (PEB)
    • Expectation Maximisation (EM)
    • Application: EEG source reconstruction

Notes and matlab code for EM example can be downloaded here: ZIP

7. Model comparison

    • Bayes factors and odds ratios
    • Model evidence for Bayesian GLMs
    • Accuracy and complexity (AIC/BIC)
    • Bias-variance decomposition
    • Application: EEG source reconstruction
    • Bayesian model averaging
    • Application: Nonlinear EEG source reconstruction

Notes can be downloaded here: PDF

8. Spectral Estimation

    • Fourier series and periodograms
    • Autocorrelation and power spectral density
    • Cross-correlation and cross spectral density
    • Coherence and Phase
    • Welch and multitaper methods
    • Localisation of MEG Gamma activity

Notes and matlab code for sunspot spectra can be downloaded here: ZIP

9. Approximate Bayesian Inference

    • Laplace approximation
    • Kullback-Liebler divergence
    • Variational Bayes and EM
    • Mixture models
    • Application: Group analysis of imaging data

Notes can be downloaded here: PDF

10. Nonlinear models

    • Central Limit Theorem
    • Independent Component Analysis
    • Application: EEG artifact removal
    • Discriminant Analysis
    • Application: Estimating perceptual state from fMRI

Notes can be downloaded here: PDF